You may hear from others with other ideas but here is mine. Since you are limited in the space you have (3-3.5 inches) to land on the sill, you are probably left with notching the joists. If you have the room to add a cleat under the full size portion where they meet the concrete foundation, all the better. This, in my mind anyway, saves you from trying to toenail all of them to the concrete. You just have to attach the cleat. Or I suppose you could use hangers, but I'm thinking of less drilling/fastening into the concrete with the cleat.
the gusset idea would be to add plywood to each side of your married joists, spanning the notched area. This will strengthen the area where the notch would make the joist more prone to cracking/failing. We havent discussed the span, but since you are opening up the whole floor, you will have the luxury of deciding if you want to work with single or double 2 bys to create you floor framing. Also, if you will have the room, a 3/4 subfloor will add some rigidity that might be lacking in the current situation also.
Just some thoughts.....